Exercise 1.1 Show how to ﬁnd A and B, given A + B and A − B. Answer To ﬁnd A simply sum them and divide by two componentwise. To ﬁnd B simply subtract them and divide by two componentwise. Deﬁne the vectors C and D:

C=A+B D=A−B Then:

A= = = = =

C+D 2 (A + B) + (A − B) 2 A+B+A−B 2 A+A+B−B 2 2A 2

=A Also:

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B= = = = =

C−D 2 (A + B) − (A − B) 2 A+B−A+B 2 A−A+B+B 2 2B 2

=B

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is deﬁned by the equation vrel = vA − vB . where vA is the velocity of A and vB is the velocity of B. Determine the velocity of A relative to B if
vA = 30 km/hr east vB = 40 km/hr north. vrel . Answer See Figure.4 The velocity of sailboat A relative to sailboat B.
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.Exercise 1.

In terms of A. A. etc.
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. B. show that the vector sum of the successive sides of the triangle (AB + BC + CA) is zero. where the side AB is from A to B. and C that extend from the origin. and C.Exercise 1.8 A triangle is deﬁned by the vertices of three vectors. Answer See Figure. B.

Find the surface swept out by the tip of r if (a) (r − a) · a = 0.3
Scalar or Dot Product
The vector r. starting at the origin. terminates at and speciﬁes the
Exercise 3. The vector a is constant (constant in magnitude and direction)
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. z).1.3
point in space (x. y.

we simply ﬁnd two that are a scalar multiple of one another: Parallel or Antiparallel P Q
They are antiparallel since the scalar multiple is negative.
(−2)P = (−2)(3i + 2j − k) = −6i − 4j + 2k =Q
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.To ﬁnd two vectors that are parallel or antiparallel. so the vectors are in opposite directions.

b. Thus. it follows that e ⊥ a and f ⊥ a. then the angle between e and f .
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. and d all lie in the same place. c. Then. It is given that e × f = 0.10 If four vectors a. ef sin θ = 0. show that (a × b) × (c × d) = 0 Hint: Consider the directions of the cross-product vectors. To have this. Answer Let e = a × b and f = c × d. we have that e and f must be either parallel or antiparallel.Exercise 4. θ must be either 0◦ or 180◦ .